Notes on Logic

Basics

Christopher L. Holland
christopher.holland@slu.edu

Saint Louis University

January 23, 2025

What Is Logic?

What Is Logic?

The study of methods for evaluating arguments

Statements and Propositions

Statement
A sentence that expresses a proposition
Proposition
The content or meaning of a (fully interpreted) statement

Statements and Propositions

Two or more statements may express the same proposition.

  • Both “the moon orbits the earth” and “the earth is orbited by the moon” express the same proposition.
  • Both “Jesus is Lord” and “Kyrios Iēsous” express the same proposition, that Jesus is Lord

Statements and Propositions

A single statement may have more than one meaning.

I shot an elephant in my pajamas

   — Groucho Marx

Truth Value

Truth Value
The property of a proposition being either true or false
  • Statements express propositions
  • Questions, proposals, suggestions, commands, and exclamations do not express propositions.

Which of the following are statements?

  • Today is Tuesday.
  • What day of the week is it?
  • Please walk the dog.
  • If it is raining.
  • Mom said “If it is raining.”
  • Keep out!
  • Wet cement.
  • Ouch!
  • Murder is illegal.
  • Murder is wrong.
  • Yuck!

Three fundamental laws of logic

  • The law of identity: \(p\) is \(p\)
  • The law of noncontradiction: it is impossible that both \(p\) and not-\(p\) are true
  • The law of excluded middle: either \(p\) is true or not-\(p\) is true

What Is an Argument?

What Is an Argument?

A set of statements where some of the statements, the premises, are used to support another statement, the conclusion.

Premise
A statement intended to support the conclusion
Conclusion
statement supported by the premise(s)

Indicator words

  • Conclusion indicator words
    • consequently, thus, therefore, it follows that, as a result, hence, so, which means that
  • Premise indicator words
    • in view of the fact, because, assuming that, since, due to the fact that, inasmuch as, for
  • Logical operators
    • not, or, and, if-then

Non-Arguments

Non-Arguments

  • Reports
  • Opinions
  • Illustrations
  • Explanations
  • Conditionals

Explanations

Example: The sky appears blue from the earth’s surface because light rays from the sun are scattered by particles in the atmosphere.

Explanans / Explicans

  • light rays from the sun are scattered by particles in the atmosphere.

Explanandum / Explicandum

  • The sky appears blue from the earth’s surface

Conditionals

If-then statements. The if part is called the antecedent. The then part is called the consequent.

  • The antecedent is sufficient condition.
  • The consequent is the necessary condition.

Conditional Variants

  • if it is raining, then the ground is wet
    if (antecedent), then (consequent)
  • the ground is wet if it is raining
    (consequent) if (antecedent)
  • given that it is raining, the ground is wet
    given that (antecedent), (consequent)
  • the ground is wet, given that it is raining
  • (consequent), given that (antecedent)
  • it is raining only if the ground is wet
    (antecedent) only if (consequent)

Conditionals

A conditional is not an argument, but many arguments include conditionals. For example:

1.
If it is raining, then the ground is wet.
2.
The ground is not wet.
∴ 3.
So, it is not raining.

Reference

Howard-Snyder, Frances, Daniel Howard-Snyder, and Ryan Wasserman. 2013. The Power of Logic. New York: McGraw-Hill.
Pojman, Louis P. 2006. Philosophy: The Pursuit of Wisdom. 5th ed. Belmont, CA: Wadsworth.